#105219

order	512
nilpotency class	3
isoclinism	isoclinic class of #105209
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

300 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,3,6,4,7,8,5)(10,13)(14,16)(17,18,21,23,22,26,20,25)(19,27,29,32,28,31,30,24);(10,14)(12,15)(13,16)(18,23)(19,28)(20,21)(24,31)(25,26)(27,32);(2,5)(3,8)(6,7)(9,10)(11,13)(12,16)(14,15)(17,19,20,29,22,28,21,30)(18,24,25,27,26,32,23,31)

Polycyclic presentation:

2^9|a^2=ei,c^2=d,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fi,e^c=eh,f^c=fgi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12415);